This talk will first present a brief review of the recent result of Holden and Sun concerning a scaling limit of discrete (random) conformal surface maps in terms of critical site percolation on uniform triangulations. We will then describe the potential relevance of this set-up for studying the action of the universal Galois group on branched representations of Riemann surfaces that we are currently exploring.

We introduce a Loewner potential, associated to a collection of n disjoint simple chords (multichords) joining 2n boundary points of a simply connected domain in the complex plane. It is first motivated by the large deviations of multiple SLE, a probabilistic model of interfaces in 2D statistical mechanics configurations: We prove a strong large deviation principle (LDP) for multiple chordal SLE0+ curves with respect to the Hausdorff metric. The rate function differs from the potential by an additive constant depending only on the boundary data, that satisfies PDEs arising as a semiclassical limit of the Belavin-Polyakov-Zamolodchikov equations in conformal field theory. Moreover, we show that the potential can be expressed intrinsically in terms of determinants of Laplacians. Furthermore, we prove that multiple SLE_0 can be defined as the unique multichord minimizing the potential in the upper half-plane for a given boundary data and show it to be the real locus of a rational function. As a by-product, we obtain an analytic proof of the Shapiro conjecture in real enumerative geometry, first proved by Eremenko and Gabrielov: if all critical points of a rational function are real, then the function is real up to post-composition by a Mobius map.

The transition semigroup \(P_t\) of a continuous-time Markov process on a metric space \(X\) acts as an operator on the space \(B(X)\) of bounded measurable function on \(X\), as well as on the space \(\mathcal{P}(X)\) of probability measures on \(X\). Certain ``smoothing'' properties of \(P_t\) can be expressed in terms of contractive inequalities on \(\mathcal{P}(X)\) involving various distances between probability measures, such as the Kantorovich--Wasserstein optimal transport distance. We study a duality relationship between such inequalities and ``reverse'' functional inequalities for \(P_t\) acting on functions, such as the reverse Poincare and reverse log Sobolev inequalities. I will discuss some applications including results about rates of convergence to equilibrium and smoothness of transition densities. This is joint work with Fabrice Baudoin (University of Connecticut). Our preprint is at https://arxiv.org/abs/2004.02050.

Why do 300 million sperm cells search for the oocyte in human fertilization when only a single sperm cell is necessary? Why do 1000 calcium ions enter a dendritic spine when only two ions are necessary to activate the relevant receptors? The seeming redundancy in these and many other systems can be understood in terms of extreme first passage time (FPT) theory. While FPT theory is often used to estimate timescales in biology, the overwhelming majority of studies focus on the time it takes a given single searcher to find a target. However, in many scenarios the more relevant timescale is the FPT of the first searcher to find a target from a large group of searchers. This fastest FPT is called an extreme FPT and is often orders of magnitude faster than the FPT of a given single searcher. In this talk, we will explain recent results in extreme FPT theory and show how they modify traditional notions of diffusion timescales.

I will define hierarchies of difference equations whose solutions, called \(\tau\)-functions, are matrix elements for the action of loop groups, \(\widehat{GL_n}\), on \(n\)-component fermionic Fock space. In the simplest case, \(n=2\), these \(\tau\)-functions are determinants of Hankel matrices, and one can apply the famous Desnanot-Jacobi identity to see that they satisfy a \(Q\)-system. \(Q\)-systems are discrete dynamical systems that appear in many areas of mathematics, so it is interesting to study the more general, \(n>2\) hierarchies. I will discuss these new hierarchies of difference equations and the progress I have made in investigating them. (The first part of this talk is based on joint work with Maarten Bergvelt.)

In 1975, McKean used a stochastic cascade to construct a solution to the KPP equation in the physical domain. In 1997, Le Jan and Sznitmann used a similar approach for the Navier-Stokes Equations (NSE) in the Fourier domain. Their construction leads to a problem known as stochastic explosion: will the cascade reach every finite horizon within finitely many steps (non-explosion), or can it happen that there will be infinitely many branches in a finite horizon (explosion)? This problem has been studied for the cascade associated with the NSE, the complex Burgers equation, and the alpha-Ricatti equation. In these equations, non-explosion is equivalent to the uniqueness of solutions, which may be resolved by an analytic approach. From a probabilistic perspective, however, the problem of stochastic explosion can be formulated generally in terms of a branching Markov chain and an independent family of holding times without connection with a PDE. We use “cutset” arguments to give a criterion for non-explosion. Applications include the KPP equation in Fourier domain, the Bessel cascade of NSE, and the Yule process. In this talk, I will give a synopsis of this approach. Joint work with Radu Dascaliuc, Enrique Thomann and Edward Waymire.

Free probability theory is introduced by Voiuclescu to attack the free group von Neumann algebras isomorphism problem. The key notion in this theory is the free independence relation which is a noncommutative analogue of independence relation in classical probability. In this talk, by generalizing the free independence relation, I will introduce a new family of noncommutative independence relations. Furthermore, I will briefly show you some analytic results and combinatorial results for these new relations.

I will talk about integrable equations that arise in the study of soft/slender elastic objects with negative Gaussian curvature. I will discuss a "topological" defect that is supported by $C^{1,1}$ solutions of hyperbolic Monge-ampere equations and is an obstruction to smoothing. I will then make connections with discrete integrable systems defined on quadgraphs and then conclude with a discussion of connections to other problems in math physics. This is joint work with Toby Shearman and Ken Yamamoto.

The Kontsevich-Zorich cocycle is a fundamental object in the study of the chaotic dynamics of the Teichmueller geodesic flow on moduli spaces of hyperbolic surfaces. A dynamical form of the Law of Large Numbers shows that the exponential growth rate of the norm of the cocycle along typical orbits converges to its space average. In this talk, we show that these growth rates also obey a Central Limit Theorem. This provides one of the first instances in which such a result holds for deterministic cocycles over flows and generalizes a long history of results in the study of random walks on groups. The main ingredient is a spectral gap result which is established via the theory of anisotropic Banach spaces. No background in Teichmueller theory or spectral theory will be assumed.

Machine learning, and in particular neural network models, have revolutionized fields such as image, text, and speech recognition. Today, many important real-world applications in these areas are driven by neural networks. There are also growing applications in finance, engineering, robotics, and medicine. Despite their immense success in practice, there is limited mathematical understanding of neural networks. Our work shows how neural networks can be studied via stochastic analysis, and develops approaches for addressing some of the technical challenges which arise. We analyze both multi-layer and one-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. In the case of single layer neural networks, we rigorously prove that the empirical distribution of the neural network parameters converges to the solution of a nonlinear partial differential equation. In addition, we rigorously prove a central limit theorem, which describes the neural network's fluctuations around its mean- field limit. The fluctuations have a Gaussian distribution and satisfy a stochastic partial differential equation. For multilayer neural networks we rigorously derive the limiting behavior of the neural networks output. We also prove convergence to the global minimum under appropriate conditions. We demonstrate the theoretical results in the study of the evolution of parameters in the well known MNIST and CIFAR10 data sets.

I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. In the linear case, the FBSDE can be explicitly solved and we recover results of Caravenna, Sun, and Zygouras. Joint work with Yu Gu (CMU).

We establish the Freidlin--Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation with multiplicative noise in one spatial dimension. That is, we introduce a small parameter $\sqrt{\epsilon}$ to the noise, and establish an LDP for the trajectory of the solution. Such a Freidlin--Wentzell LDP gives the short-time, one-point LDP for the KPZ equation in terms of a variational problem. Analyzing this variational problem under the narrow wedge initial data, we prove a quadratic law for the near-center tail and a 5/2 law for the deep lower tail. These power laws confirm existing physics predictions in the literature. We will also discuss a limit shape problem which arises from the deep lower tail conditioning. Joint work with Li-Cheng Tsai.